Lawson criterion

In nuclear fusion research, the Lawson criterion, first derived on fusion reactors (initially classified) by John D. Lawson in 1955 and published in 1957,[1][dead link] is a general system measure that defines the conditions needed for a fusion reactor to reach ignition, that is, that the heating of the plasma by the products of the fusion reactions is sufficient to maintain the temperature of the plasma against all losses without external power input. As originally formulated the Lawson criterion gives a minimum required value for the product of the plasma (electron) density ne and the "energy confinement time" τE{\displaystyle \tau _{E}}.

Later analysis suggested that a more useful figure of merit is the "triple product" of density, confinement time, and plasma temperature T. The triple product also has a minimum required value, and the name "Lawson criterion" may refer to this inequality.

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The central concept of the Lawson criterion is the energy balance for any fusion power plant, using a hot plasma. This is shown below:

Net Power = Efficiency × (Fusion − Radiation Loss − Conduction Loss)

Net Power is the excess power beyond that needed internally for the process to proceed in any fusion power plant.

Efficiency how much energy is needed to drive the device and how well it collects power.

Fusion is rate of energy generated by the fusion reactions.

Radiation is the energy lost as light, leaving the plasma.

Conduction is the energy lost, as mass leaves the plasma.

Lawson calculates the fusion rate by assuming that any fusion reactor contains a hot plasma cloud which has a Gaussian curve of energy. Based on that assumption, he estimates the first term, the fusion energy coming from a hot cloud using the volumetric fusion equation.[2]

Fusion = Number Density of Fuel A × Number Density of Fuel B × Cross Section(Temperature) × Energy Per Reaction

Fusion is the rate of fusion energy produced by the plasma

Number density (particles per unit volume) is the density of the respective fuels (or just one fuel, in some cases).

Cross Section is a measure of the probability of a fusion event, based on plasma temperature

Energy per reaction is the energy made in each fusion reaction

This equation is typically averaged over a population of ions which has a normal distribution. For his analysis, Lawson ignores conduction losses. In reality this is nearly impossible, practically all systems lose energy through mass leaving. Lawson then estimated[2] the radiation losses using the equation below.

When applied to the fusor Lawson's analysis is used as an argument that conduction and radiation losses are the key impediment to reaching net power. Fusors use a voltage drop to accelerate and collide ions, resulting in fusion.[4] The voltage drop is generated by wire cages, and these cages conduct away particles. Polywell are improvements on this design, designed to reduce conduction losses by removing the wire cages which cause them.[5] Regardless, it is argued that radiation is still a major impediment.[6]

The confinement timeτE{\displaystyle \tau _{E}} measures the rate at which a system loses energy to its environment. It is the energy density W{\displaystyle W} (energy content per unit volume) divided by the power loss density Ploss{\displaystyle P_{\mathrm {loss} }} (rate of energy loss per unit volume):

τE=WPloss{\displaystyle \tau _{E}={\frac {W}{P_{\mathrm {loss} }}}}

For a fusion reactor to operate in steady state, the fusion plasma must be maintained at a constant temperature. Thermal energy must therefore be added to it (either directly by the fusion products or by recirculating some of the electricity generated by the reactor) at the same rate the plasma loses energy. The plasma loses energy through mass (conduction loss) or light (radiation loss) leaving the chamber.

For illustration, the Lawson criterion for the deuterium–tritium reaction will be derived here, but the same principle can be applied to other fusion fuels. It will also be assumed that all species have the same temperature, that there are no ions present other than fuel ions (no impurities and no helium ash), and that deuterium and tritium are present in the optimal 50-50 mixture.[7] Ion density then equals electron density and the energy density of both electrons and ions together is given by

W=3nkBT{\displaystyle W=3nk_{\mathrm {B} }T}

where kB{\displaystyle k_{\mathrm {B} }} is the Boltzmann constant and n{\displaystyle n} is the particle density.

The volume ratef{\displaystyle f} (reactions per volume per time) of fusion reactions is

The volume rate of heating by fusion is f{\displaystyle f} times Ech{\displaystyle E_{\mathrm {ch} }}, the energy of the charged fusion products (the neutrons cannot help to heat the plasma). In the case of the deuterium–tritium reaction, Ech=3.5MeV{\displaystyle E_{\mathrm {ch} }=3.5\,\mathrm {MeV} }.

The Lawson criterion, or minimum value of (electron density * energy confinement time) required for self-heating, for three fusion reactions. For DT, nτE minimizes near the temperature 25 keV (300 million kelvins).

The quantity T/⟨σv⟩{\displaystyle T/\langle \sigma v\rangle } is a function of temperature with an absolute minimum. Replacing the function with its minimum value provides an absolute lower limit for the product nτE{\displaystyle n\tau _{E}}. This is the Lawson criterion.

A still more useful figure of merit is the "triple product" of density, temperature, and confinement time, nTτE. For most confinement concepts, whether inertial, mirror, or toroidal confinement, the density and temperature can be varied over a fairly wide range, but the maximum attainable pressure p is a constant. When such is the case, the fusion power density is proportional to p2<σv>/T 2. The maximum fusion power available from a given machine is therefore reached at the temperature T where <σv>/T 2 is a maximum. By continuation of the above derivation, the following inequality is readily obtained:

The quantity T2⟨σv⟩{\displaystyle {\frac {T^{2}}{\langle \sigma v\rangle }}} is also a function of temperature with an absolute minimum at a slightly lower temperature than T⟨σv⟩{\displaystyle {\frac {T}{\langle \sigma v\rangle }}}.

For the deuterium–tritium reaction, the minimum of the triple product occurs at T = 14 keV. The average <σv> in this temperature region can be approximated as[8]

This number has not yet been achieved in any reactor, although the latest generations of machines have come close. JT-60 reported 1.53x1021 keV.s.m−3.[9] For instance, the TFTR has achieved the densities and energy lifetimes needed to achieve Lawson at the temperatures it can create, but it cannot create those temperatures at the same time. ITER aims to do both.

As for tokamaks there is a special motivation for using the triple product. Empirically, the energy confinement time τE is found to be nearly proportional to n1/3/P 2/3. In an ignited plasma near the optimum temperature, the heating power P equals fusion power and therefore is proportional to n2T 2. The triple product scales as

Satisfaction of this criterion at the density of solid deuterium–tritium (0.2 g/cm³) would require a laser pulse of implausibly large energy. Assuming the energy required scales with the mass of the fusion plasma (Elaser ~ ρR3 ~ ρ−2), compressing the fuel to 103 or 104 times solid density would reduce the energy required by a factor of 106 or 108, bringing it into a realistic range. With a compression by 103, the compressed density will be 200 g/cm³, and the compressed radius can be as small as 0.05 mm. The radius of the fuel before compression would be 0.5 mm. The initial pellet will be perhaps twice as large since most of the mass will be ablated during the compression.

The fusion power density is a good figure of merit to determine the optimum temperature for magnetic confinement, but for inertial confinement the fractional burn-up of the fuel is probably more useful. The burn-up should be proportional to the specific reaction rate (n2<σv>) times the confinement time (which scales as T -1/2) divided by the particle density n:

^It is straightforward to relax these assumptions. The most difficult question is how to define n{\displaystyle n} when the ion and electrons differ in density and temperature. Considering that this is a calculation of energy production and loss by ions, and that any plasma confinement concept must contain the pressure forces of the plasma, it seems appropriate to define the effective (electron) density n{\displaystyle n} through the (total) pressure p{\displaystyle p} as n=p/2Ti{\displaystyle n=p/2T_{\mathrm {i} }}. The factor of 2{\displaystyle 2} is included because n{\displaystyle n} usually refers to the density of the electrons alone, but p{\displaystyle p} here refers to the total pressure. Given two species with ion densities n1,2{\displaystyle n_{1,2}}, atomic numbers Z1,2{\displaystyle Z_{1,2}}, ion temperature Ti{\displaystyle T_{\mathrm {i} }}, and electron temperature Te{\displaystyle T_{\mathrm {e} }}, it is easy to show that the fusion power is maximized by a fuel mix given by n1/n2=(1+Z2Te/Ti)/(1+Z1Te/Ti){\displaystyle n_{1}/n_{2}=(1+Z_{2}T_{\mathrm {e} }/T_{\mathrm {i} })/(1+Z_{1}T_{\mathrm {e} }/T_{\mathrm {i} })}. The values for nτ{\displaystyle n\tau }, nTτ{\displaystyle nT\tau }, and the power density must be multiplied by the factor (1+Z1Te/Ti)⋅(1+Z2Te/Ti)/4{\displaystyle (1+Z_{1}T_{\mathrm {e} }/T_{\mathrm {i} })\cdot (1+Z_{2}T_{\mathrm {e} }/T_{\mathrm {i} })/4}. For example, with protons and boron (Z=5{\displaystyle Z=5}) as fuel, another factor of 3{\displaystyle 3} must be included in the formulas. On the other hand, for cold electrons, the formulas must all be divided by 4{\displaystyle 4} (with no additional factor for Z>1{\displaystyle Z>1}).